Counting irreducible polynomials with prescribed coefficients over a finite field

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by equivalence classes of polynomials with prescribed coefficients. Simplified expressions are derived for some special cases. Our results extend some earlier results.     

Entire functions that share one value with their linear differential polynomials

This paper studies the uniqueness problem on entire function that share a finite, nonzero value with their linear differential polynomials and proves some theorems which generalize some results given by Jank, Mues and Volkmann, P. Li, J.P. Wang and H.X. Yi.     

SOME REMARKS ON QUASI-DIFFERENTIAL EXPRESSIONS WITH MATRIX-VALUED COEFFICIENTS

Abstract

This paper is Concerned with the relationship between ordinary linear quasi-differential expressions, defined in terms of locally Lebesgue integrable matrix coefficients given on an interval of the real line and “Classical” differential expressions with smooth (differentiable to certain prescribed orders) matrix coefficients. This relationship wan investigated in a recent paper of Everitt and Race [1] for the case of scalar quasi-differential expressions of Shin-Zettl type. The present work extends the ideas given there, to the more general quasi-differential expressions considered by Frentzen in recent years (see, for example [4,5]) and applies them to products and polynomials of expressions.     

On the inversion of certain moment matrices

An explicit representation of the elements of the inverses of certain patterned matrices involving the moments of nonnegative weight functions is derived in this paper. It is shown that a sequence of monic orthogonal polynomials can be generated from a given weight function in terms of Hankel-type determinants and that the corresponding matrix inverse can be expressed in terms of their associated coefficients and orthogonality factors. This result enables one to obtain an explicit representation of a certain type of approximants which apply to a wide class of positive continuous functions. Convenient expressions for the coefficients of standard classical orthogonal polynomials such as Legendre, Jacobi, Laguerre and Hermite polynomials are also provided. Several examples illustrate the results.     

Irreducibility Results for Compositions of Polynomials with Integer Coefficients

We obtain explicit upper bounds for the number of irreducible factors for a class of polynomials of the form f ○ g, where f,g are polynomials with integer coefficients, in terms of the prime factorization of the leading coefficients of f and g, the degrees of f and g, and the size of coefficients of f. In particular, some irreducibility results are given for this class of compositions of polynomials.     

Irreducibility Results for Compositions of Polynomials with Integer Coefficients

We obtain explicit upper bounds for the number of irreducible factors for a class of polynomials of the form f ○ g, where f,g are polynomials with integer coefficients, in terms of the prime factorization of the leading coefficients of f and g, the degrees of f and g, and the size of coefficients of f. In particular, some irreducibility results are given for this class of compositions of polynomials.     

On condition numbers of polynomial eigenvalue problems

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.     

On the enumeration of polynomials with prescribed factorization pattern

We use generating functions over group rings to count polynomials over finite fields with the first few coefficients and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic n-smooth polynomials of degree m over a finite field, as well as the number of monic n-smooth polynomials of degree m with the prescribed trace coefficient.     

Instability indices for matrix polynomials

There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directions for the matrix coefficients which are Hermitian to the total number of unstable eigenvalues for the polynomial. Herein we extend the theory to ?-even matrix polynomials of any finite degree. In particular, unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two. We also consider Hermitian matrix polynomials, and derive an index which counts the number of eigenvalues with nonpositive imaginary part. The results are refined if we consider the Hermitian matrix polynomial to be a perturbation of a ?-even polynomials; however, this refinement requires additional assumptions on the matrix coefficients.     

Zero distribution of random polynomials

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that under mild assumptions on the coefficients, their zeros are asymptotically uniformly distributed near the unit circumference. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane and quantify this convergence. In our results, random coefficients may be dependent and need not have identical distributions.     

Littlewood–Richardson polynomials

We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood–Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author. The new rule provides a formula for these polynomials which is positive in the sense of W. Graham. We apply this formula for the calculation of the product of equivariant Schubert classes on Grassmannians which implies a stability property of the structure coefficients. The first manifestly positive formula for such an expansion was given by A. Knutson and T. Tao by using combinatorics of puzzles while the stability property was not apparent from that formula. We also use the Littlewood–Richardson polynomials to describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by A. Okounkov and G. Olshanski.     

Polynomials with integer coefficients and their zeros

We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.     

Padé approximation and generalized moments

We propose studying generalized moment representations of a form in which it suffices to apply a system of orthogonal polynomials in order to procure the biorthogonality conditions in the construction of superdiagonal Padé polynomials using generalized moment representations. The algebraic polynomials in the moment representation are to be sought as the linear forms of biorthogonal polynomials. We obtain the relations between the coefficients of these linear forms and the generalized moments, and we also establish conditions for the existence and uniqueness of generalized moment representations of polynomial form. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 110–115.     

Gale duality bounds for roots of polynomials with nonnegative coefficients

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.     

A note on Koornwinder's polynomials with weight function (1−x)α(1+x)β+Mδ(x+1)+Nδ(x−1)

We derive the distribution and three-term recurrence relation for the Koornwinder [2] polynomials with weight function shown above using the method developed in our paper [3]. We are able to obtain explicit expressions for the linear functional in terms of the coefficients of the three-term recurrence relation for the Jacobi polynomials and we obtain the distribution using a more direct approach.     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

On shifted Eisenstein polynomials

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree \(n\) polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.     

On the generalized Ball bases

The Ball basis was introduced for cubic polynomials by Ball, and two different generalizations for higher degree m polynomials have been called the Said–Ball and the Wang–Ball basis, respectively. In this paper, we analyze some shape preserving and stability properties of these bases. We prove that the Wang–Ball basis is strictly monotonicity preserving for all m. However, it is not geometrically convexity preserving and is not totally positive for m>3, in contrast with the Said–Ball basis. We prove that the Said–Ball basis is better conditioned than the Wang–Ball basis and we include a stable conversion between both generalized Ball bases. The Wang–Ball basis has an evaluation algorithm with linear complexity. We perform an error analysis of the evaluation algorithms of both bases and compare them with other algorithms for polynomial evaluation.     

Computation of Gauss-type quadrature formulas

Gaussian formulas for a linear functional L (such as a weighted integral) are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to L. In Gauss-type formulas, one or more extraneous conditions (such as pre-assigning certain nodes) replace some of the equations expressing exactness when applied to high-order polynomials. These extraneous conditions may be applied by modifying the same number of recursion coefficients. We survey the methods of computing formulas from recursion coefficients, methods of obtaining recursion coefficients and modifying them for Gauss-type formulas, and questions of existence and numerical accuracy associated with those computations.     

Unimodularity of zeros of self-inversive polynomials

We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang as well as by Lalín and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lalín and Rogers on computational evidence.